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ON SCOPE and C-COMMAND 1. Introduction C-Command and Scope

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ON SCOPE and C-COMMAND 1. Introduction C-Command and Scope ON SCOPE AND C-COMMAND MARCUS KRACHT 1. Introduction C-command and scope are notions used to talk about readings of a given sentence. In the sentence (1) Every man loves a woman. one says that in one reading every (or every man) “takes scope over” a (or a woman), and that in the other reading it is a that “takes scope over” every. Interchangeably, one says that in the first sentence every man (however, arguably not every) c-commands a woman, and that the situation is reversed in the second reading. To have different readings is prima facie a matter of interpretation, or meaning, as I will consistently say. However, both scope and c- command happen to be defined in terms of structure. Scope is typically based on the string, while c-command is based on the tree structure that the sentence is thought to have. In what is to follow I shall be concerned with the exact ways in which tree based notions reflect string based ones, and vice versa; and with the question of how it is exactly that c-command and scope bear on interpretation. More precisely, I shall ask to what extent the c-command relation is sufficient in nail- ing down the structure inasfar as it matters for interpretive purposes. And thirdly I shall ask whether the different definitions given in the literature will make be different in this. There is a connection between c-command and scope that has its roots both in the intellectual history of the notions themselves and the use that is made of them in the technical literature. Unfortunately, for neither notion is there an agreed definition. In this note I shall review some definitions and show that for all intents and purposes these definitions do give identical results. 2. Conventions We assume that languages are formed over a basic set which we call alphabet. Elements of the alphabet are called letters. From let- ters we form strings by concatenation. The symbol of concatenation 1 2 englishMARCUS KRACHT is a. Actual letters or strings are written using typewriter font, like this: Pxy→Qy. Other symbols are just proxy for strings of a certain kind. Typically, I use ~x, ~u to denote strings, but other wuthors have different habits. Concatenation is sometimes not even written. I write interchangeably (2) abbac and (3) aababaaac Natural languages work a little differently. The letters of the language are actually the words, and the alphabet therefore should be called a dictionary. The fact that the words are themselves composed using letters of a different alphabet (unlike Chinese) may give rise to what is known as segmentation problems. These are problems to identify the basic words in a string. Suppose, by way of example, that we have a binary relation symbol P and a unary relation symbol Pa in addition to a constant a and a variable x. Then Pax is segmentable into a succession of Pa and x as well as a sequence of P, a and x. The presence of a blank eliminates most segmentation problems in written language. However, for the purpose of this note we assume that the language of the simple kind above: the basic constituents (alias words) are simply the letters of the alphabet. 3. Logic The notion of scope is technically used only in connection with quan- tification. It is thus not even defined what the scope of a connective like → or ∨ is. The definitions of scope of a quantifier vary. [5] says the following. First, it is agreed that if A is a well-formed formulae and y a variable then (y)A is a well-formed formula as well. In ((y)A,“A” is the scope of the quantifier (y). To understand this, one has to be reminded of the fact that in logic one first specifies an alphabet of symbols and then a set of so called well-formed formulae (= ‘wffs’). Additional sets, such as the sets of terms, variables and so on are also defined. The definition typically amounts to a context-free grammar, where the sets are replaced by nonterminals standing in for these sets. Thus, if Wff is a nonterminal denoting the set of wfs, and Var a nonterminal for the set of variables, it is first agreed that we have a rule of the form (4) Wff → ‘(’ Var ’)’ Wff englishON SCOPE AND C-COMMAND 3 To say this is tantamount to saying that if ~x is in Var and ~y in Wff then the concatenation of ‘(’ then ~x then ‘)’ and then ~y is in Wff as well. It is customary to write this concatenation either as (5) (a~xa)a~y thus using the a as a symbol of concatenation, or simply—suppressing this symbol—as (6) (~x)~y Notice that there is no need for quotes anymore: we concatenate only strings! Upon this one proceeds right away to the specification of meaning. Strings belonging to different sets will denote different things; in pred- icate logic, variables denote objects (under a given assignment), so do constants. Unary predicates denote sets of objects, binary predicates letters denote relations between objects. Finally, wffs denote truth val- ues: these are true and false. We generally do not say, however, that an expression denotes true under a valuation but that it is true. What the definition (4) provides is a way to determine whether the expression formed using the syntactic rule is true or not. Thus, given we know the meaning of ~x and ~y, we should be able to say what (~x)~y means. The definition (not shown here) is effectively the one standardly given. I paraphrase [5] as follows. (y)A is true under a given assignment σ if A is true under every assignment σ0 which is different from σ at most in the value of y. The notion of scope is not needed here; but we notice that Mendelsohn declares the scope of (y) to be A. Scope is not even needed to talk about free and bound occurrences of a variable. Here we are told that an occurrence of a variable x is bound either if it is in an expression of the form (x) or in the scope of such an expression. Machover in [4] gives a different account. He issues the following clause: (4) If x is a variable and b is an L-formula then ∀xb (the string obtained by concatenating a single occurrence of ∀, a single occurrence of x and the string b, in this order) is an L-formula. Notice that this defines an altogether different language even if the same underlying letters were used! This is because (~x) is a different string than is ∀~x. Mendelsohn uses y to denote a string that is a variable, while Machover uses the symbol x. We use ~x to say the same. 4 englishMARCUS KRACHT These differences are just stylistic; however, in Mendelsohn’s language the universal quantifier is denoted by a pair of brackets enclosing the string, while in Machover’s it is the symbol ∀ and no brackets are used. (4) leaves nothing to desire in terms of explicitness. Greek letters are used as variables for strings that are L-formulae, while x is a variable over strings which are variables (of the language). Then we are told that A formula ∀xb constructed according to (4) is called a universal formula; here x is the variable of quantification and the string xb is the scope of the initial occurrence of the universal quantifier. Here the scope includes every bound occurrence of the variable, which was not the case in the previous definition. This definition fixes a few question of detail that the previous defi- nition left implicit. Machover distinguishes between strings and occur- rences of strings. We can make this distinction formal as follows. Let ~x be a string. An occurrence is a pair of strings. We say that h~v, ~wi is an occurrence of ~x in ~z iff ~z = ~va~xa ~w. Thus, the notion of an occur- rence makes sense only when we are talking about a given string. Say that string is aabba. In this string, a has three occurrences: hε, abbai (where ε denotes the empty string), ha, bbai and haabb, εi. b has only two occurrences: haa, bai and haab, ai. If you dislike the complica- tion about pairs, think of an occurrence as a string with some portion underlined, such as this: (7) qwertyuiop This gives you both the host string (qwertyuiop) and the substring (rty). The occurrence is obtained by naming the part that is to the left of the line (qwe) and the part that is to the right (uiop). To see that this care is needed let us look at the following formula: (8) ∃xPx→∃x∀yxQy In this string, the quantifier ∃x has two occurrences, so does the vari- able x: it occurs 4 times. We have to be exact as to which quantifier occurrence binds which variable occurrence. Talk of “the quantifier ∃x” and “the variable x” is not enough in this circumstance. Thus, in the definition given by Mendelsohn, he ought to have said this: In (y)A,“A” is the scope of the leftmost occurrence of the quantifier (y). As scopes are assumed to be disjoint from their quantifiers, this may be judged an innocent matter. For the only occurrence of (y) that is englishON SCOPE AND C-COMMAND 5 not in A is already the leftmost one. However, and this is mostly left implicit, the scope of (y) remains A even when we embed the formula into a larger one.
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